The Hausdorff dimension of visible sets of planar continua

• Autores: Toby C. O'Neil
• Localización: Transactions of the American Mathematical Society, ISSN 0002-9947, Vol. 359, Nº 11, 2007, págs. 5141-5170
• Idioma: inglés
• DOI: 10.1090/s0002-9947-07-04460-1
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• Resumen

  • For a compact set and a point , we define the visible part of from to be the set (Here denotes the closed line segment joining to .) In this paper, we use energies to show that if is a compact connected set of Hausdorff dimension greater than one, then for (Lebesgue) almost every point , the Hausdorff dimension of is strictly less than the Hausdorff dimension of . In fact, for almost every , We also give an estimate of the Hausdorff dimension of those points where the visible set has dimension greater than for some .